All Complex Zeros of the Riemann Zeta Function Are on the Critical Line: Two Proofs of the Riemann Hypothesis

Abstract

In this paper, I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta-Function is the critical line given by ℜ(s)=1/2. Results of this paper are based on well-known theorems on the number of zeros for complex value functions (Jensen’s, Titchmarsh’s, Rouche’s theorem), with the Riemann Mapping Theorem acting as a bridge between the Unit Disk on the complex plane and the critical strip. By primarily relying on well-known theorems of complex analysis our approach makes this paper accessible to a relatively wide audience permitting a fast check of its validity. Both proofs do not use any functional equation of the Riemann Zeta-Function, except leveraging its implied symmetry for non-trivial zeroes on the critical strip.